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Nonlinear Analysis of Sampled-data Control Systems
NONLINEAR ANALYSIS OF SAMPLED-DATA CONTROL SYSTEMS...
14th World Congress ofIFAC
E-2c-13-4
Copyright © 1999 IF AC 14th Triennial World Congress, Beijing, P.R. China
Nonlinear Analysis of Sampled-data Sy~tems a
a
Co~trol
bKazuyuki Aihara
Luonan Chen
Osaka Sangyo University,3-1-1 Nakagaito, Daito, Osaka 574-8530,Japan.
Fax:81-798-70-8189;E-mail:[email protected] bThe University of Tokyo., 7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656,Japan
Abstract: This paper treats a digital-controlled Control system with both continuous-time and discretetime variables as a model of differential- difference-algebraic equations (DDA)~ presents a fWldamental analysing method of such a nonlinear DDA for local sampling, asymptotical stability, singular perturbations and bifurcations, and further shows that there exist five types of generic bifurcations at the equilibria in contrast to two types in continuous-time dynamical systems and three types in discrete-time
dynamical systems. Finally the theoretical results are applied to digital control of power systems. Numerical simulations have verified that our results are usefuL Copyright© 1999IFAC
Keywords: asymptotic stability, singular perturbations, sampled-data control, nonlineax analysis, nonlineax theory, power s~stem, bifurcation
1
Definition of DDA
2
Introduction
Many physical systems with digital devices require
mathematical descriptions not only with ordinary
The following notation is used throughout the pa-
per. Let
ference equations. As an example, for the power system with digital controllers, the dynamics of the generators as well as their continuous-time
controllers and the load dynamics together define the ordinary differential equations while algebraic
equalities are defined by the power balance equations of the transmission network[3]. On the other hand, the difference equations describe the dynamics of the digital devices. Therefore., the digital control systems of power grids are typical models mixed with both continuous and discrete time sequences which can mathematically be formulated as differential- difference-algebraic equations (DDA). In this paper, we define the constrained sampled-
data system as a model of DDA. So far, both continuous-time and discrete-time nonlinear systems have attracted considerable attention and a variety of techniques have been developed[3, 7, 5]. This paper aims at analysing the asymptotical sta-
bility and bifurcations of the DDA (5, 6] and further applying the theoretical results to digital control of
power systems. We first define a DDA model and two types of equilibria, namely quasi-equilibria and
strict equilibria due to its mixing nature of continuous and discrete time, and summarize the recent
results for locally sampling, asymptotical stability and singular perturbations of DDA[5, 6]. Then this paper shows that there exist five types of generic bifurcations at the strict equilibria in contrast to t,vo types at an equilibrium in continuous-time dynamical systems and three types at a fixed point in
discrete-time dynamical systems, and provides their bifurcation conditions for the DDA. Furthermore, this paper indicates that most of locally nondegenerate bifurcations for a DDA at strict equilibria are related with the local bifurcations of the sampling instants under some generic assumptions. Finally, the theoretical results are applied to the evaluations and designs of digital control for power systemsA
= f(x(t), x n , Yn, z(t), Zn, 0)
(1)
a) h(x(t),xn,Yn,z(t),zn,a) = 0
(2) (3)
x(t)
differential-algebraic equations, but also with dif-
Yn+l = g(x n , Yn'
Zn,
(nr:5 t < (n + l)r;n = 0,1,2, ...) be the differential-difFerence-algebraic equations (DDA) under study, where vectors x, X n , Yn,
z
and
Zn
belong
to
R,n z , Rnf,t; , Rn lJ , Rn;& and
the
Euclidean spaces
nn~, ~pectivelYA T
is a
sampling interval (non-negative real number), and 0: E R is a parameter. Define X n = x(nT),Zn ==' z(nr) and Yn is the value at the instant ni, where X n , Un and Zn are constantly hold during nT :5 t < (n + l)r for eqns4(1) and (3). Then eqns.(1)-(3) are an ordinary differential - difference - algebraic equation model (DDA). Let !:r. = af(x~x8:1'l,Z1zn) and
1/-:£1
is the determinant of /z.
11·11
is the usual
Euclidean norm of a vector or the induced matrix nonn of a matrix. E is an identity matrix. eA is the exponential fllllction of a matrix A. In this paper, a non-equality is an equation with the form of u # 0 while an inequality is the form of u < 0 or u > O. On the· other hand, an equality is an equation with the form of'U == O. x{t) is continuous due to eqn.(l) while z(t) is generally discontinuous at t == nT, n = 0, 1 1 2, .. +' namely limt_nT-O x(t) = x(nT) == X n = limt---+nr+O x(t) and limt-nr-o z(t) #- z(nT) == Zn = limt---+n.+o z(t), as far as 1hz 1 =I- 0 and 1hz + hz~J i- O. In this paper, sampling instants mean t == nT while intersampling instants imply nT < t < (n + l)T~ where n == 0,1,2, Definition 2.1 Vectors p == (x,x,Yn~ z, z) are called a strict equilibrium of DDA eqns. (1 )-(3) if /(x,xn,Yn)z,zn,a;) == 0, h(x,xn,Yn,z,zn,O:) == 0 .'Aa
and
Yn == g(xn,Yn,zn,a).
Next, without confusion, t is occasionally dropped from the related variables, e.g., xCi) or z(t) is simply expressed by x or z +
2391
Copyright 1999 IFAC
ISBN: 0 08 043248 4
NONLINEAR ANALYSIS OF SAMPLED-DATA CONTROL SYSTEMS...
14th World Congress of IFAC
Local Sampling and Stability
3
Let p == (x n, Xn,Yn, zn' Zn) satisfy eqn,,(3) and Jhz + h Zn Ij =1= O~ Then, ~cord.ing ~o the implicit function theorem there eXIsts an unique Coo mapping in open neighborhoods of p such that at sampling instants (4)
C ~
-
p
z(t) = q(x(t), X n , Yn) nT ~ t < (n + l}T (5) where qn(xn , Yn) == q(x(nT), Xn , Yn) due to Zn == limt--+n7"+O z(t). Thus the reduced. system of eqns.(l) and (3) at the neighbour of p becomes
== I(x, X n , Yn, q(:z:, X n , y.",), qn(xn~ Un))
Yn+l == 9R(X n , Yn)
=g{x n , Yn,
qn(X n , Yn))
1hz I #
0, 1h z
(9)
+ h zn I:/;O at p.
strict-equilibrium of DDA eqns. (1)-(3)"
and further the reduced system of eqns.(2) and (3) at the neighbour of p becomes forn=O,l,~u
-
where GX = 9 X n - gz.n (h z + h ZtlJ- 1 (h x + h:c n ) and GY == 9Yn - 9zn.(h% + hZn)-lhyn.. ~en we h~ve the following theorem for the stabihty of a stnct equilibrium p for all of instants. Theorem. 8.2 Assume that p= (x,x,f),z,z) is a strict equilibrium of DDA eqns. (1)-(9), and 1hz Jp =f; 0, 1hz + h Zn 1p =I- 0, lA) # O. 1. If all the eigenvalues of J at 15 have moduli less than 1 J then p is an asymptotically stable
such that at the intersampling in-
stants
x :::: fR(X t xn,Yn)
(8)
_
hz..,.(h% + hzn)-lhvn ] JZn (h z + h Zn )-lhvn
Assume IAI =F 0 and Define a J acobian
Furthermore, let p satisfy eqn.(3) and Ihzf.,; "# o. Then by using the implicit function theorem again, there exists an unique COCJ mapping in open neighborhoods of
-IZn{hz;+h:",)-l(hx+h zti )
Ivn. - Jz h;-l[hYn
2~
(6)
If anyone of the eigenvalues of J at p has modulus more than 1, then p is an unstable strictequilibrium of DDA eqns.(1)-(9).
Actuallr, we can prove that Theorem 3. 2 still holds even if IAt = o.
Let ~&, ~8 ~ and ~ are partial derivatives Z X n UYn of !R(X,Xn,Yn) at (x,Xn,Yn) with respect to (x, X n , Yn), respectively~ Then we obtain a locally sampling theorem [5, 6]. Theorem 3. 1 Assume that p = (x, x, y, z, z) is an arbitrary point which satisfies eqn~ (8). Assume that the partial derivatives of f unto appropriate
4
Singular Perturbation
For the DDA eqns.(1)-(3), we define an associated singularly perturbed system aB x(t) =
f(x(t) , X n , Yn, z(t), zn)
Yn+l =
g(xn,Yn,zn)
(10) (11) (12)
order exist, and 1hz I # 0, 1h z + h zn ~ #- 0 at any €z(t) = h(x(t), X n , Yn, z(t), Zn) solution of eqns~(l) and (3) starting from {J. Then (nr ~ t < (n + l)-r; n = 0, 1, ...) the discrete-time system for eqns. (1) and (8) can be where € is a sufficiently small positive number. That e.xpanded in a neighbourhood oip for (Xn,Yn), is, the algebraic eqn~(3) corresponds to a limit of the 1 2 (~) X n +l ~ rP7+Dxy tPT ~XY+2Dxy.p'T' Axy·~xy+... n ~ 0,1, ... fast dynami~ eqn.(12). Obviously, eqn.(12) will ap-
=
where" 4!.r:.
fT(x, Yl
x + J: iRdt
=
fR(rPt,X, YJ an rPo = X. ~%!I = [ : : : ~ ] . Dxy40T = [~. 2
[~2;c\T ~
_
D~'YtPT-
tin
~ = 8x-n. r
T
Jo
a~: tin ] ~
JR
proach the algebraic manifold when
.
positive)" Furthermore, assume that no eigenvalue of J at p has the modvlus 1, and 1h z + h Zn I" i:-
eJo'" ~dt + Jor eJ~'" *"dt!!lE.dt, 8xn. T
[}X
T
8x
0, IAl p # Ow Then there exists f > 0 such that for all positive £ < ~, stability of eqns. (10)- (12) at p is identical to that of eqns~ (1)-(9).
82~T
rT eft ~dt[82l,(!!.!l!t.-)2+
Jo
8x""
n
+~~+ 8x8z'f'I, 8z a
~dt(a2ir!!ft!!i!J.. + 8x 8y". 8x ~~+~E!&+~)dt ~ = fJx8Yn 8x n 8z8xn 8yn. 8x n 8Yn ' 8Ynoxn Ox""
~ ==
2
8 t""R 8
'
8x n 8Yn
eh
T
fr Jo
8x8xn. 8Yn
+
2
2
8 fR f!..P..t. +_El jR)dt
tJYn8x OX n
~,
8 2
_
-
- Neimark-Sacker and singularity-2 induced bifurcations at the sampling instants, in codimension one
fT '!JJ1. at 2 ~ 2 ~ 2 ~ 2J for eJt 8X (~(U::)2+~~+~M;:;+~)dt. Jo 8a; tlfl ~ Un lIn Un x '!In 811n.
Theorem 3. 1· ensures that the discretized system can locally approximate the original continuous system "at any order at sampling instants. At a strict equilibrium p, if h z and h z + h zn are nonsingular A B
==
=
Ix
-Izh~lhx
J~n - fzh;l{h xn - h%n (h.
parametric space.
Zn
Sampling instants
5.1
For the sampling instants, eqns.(1)-(3) at the neigh= (:15, X, y, z, z, et) can be expressed as bourhood of a strict equilibrium p
(7)
+h
Local Bifurcations
We will show that there are five types of bifurcations for a strict equilibrilllll tlllder certain generic conditions, Le., singularity-l induced bifurcations at the intersampling instants, and fold, flip,
+
ex; &ij;;
8%
+~~
5
n
r-r eft'T ~dt(82jf~~
J0
O. It is ev-
Theorem 4. 1 Assume that 15 = (x, x, y, z, z) are a strict equilibrium of the DDA eqns. (1)-(3), and all real parts 0/ eigenvalues for hztp are negative (or
81J n
:::tn.
E ~
ident that eqns.(10)-(12) has the same equilibrium as eqns.(1)-(3) as far as h = O.
==
~l,
eh'" *t-dt'lla ay:;; dt )
,p~R]dt
with
)-lhzn ]
.:t:'n+l "n.+l
= =
,pT
+
Dxya~7" A xll Ot.
+
~D~yo4',.- L:.;rya: - .a.:rg~ (n=D.l •... ).
9(:;Cn.Yn~qn(=nIYn.Q),Q)
+ .. ,
(13) (14)
2392
Copyright 1999 IFAC
ISBN: 0 08 043248 4
NONLINEAR ANALYSIS OF SAMPLED-DATA CONTROL SYSTEMS...
T
where
cPr =
!R(X, x,
y, a)
+
If
J; fR(cPt,x,y,a)dt
and ~zya:
=
Xn [
x+
==
-x ]
Yn - ~
,when
Q-O:'
Jh z Jp i= 0 and Jh z + hzn)p I- O. Since a strict equilibrium of the DDA eqns.(1)-(3) is a fixed point of discrete time system eqns.(13)(14), the bifurcations of eqns.. (1)-(3) at the sampling instants for the strict equilibria are basically
14th World Congress ofIFAC
Suppose that the full Jacobian matrix Ix + fX n
the same as local bifurcations of the differencealgebraic model eqns.(13)-(14). Next we examine the fixed point bifurcations of eqns.. (13)-(14). We first swnmarize the fold, flip and NeimarkSacker bifurcations [1], and then present the existence of singularity-2 induced bifurcation as well as their conditions.
Theorem 5. 1 Suppose that p = (x, x, y, z, z, a) is a strict equilibrium of eqns.(1)-(3) and 1h z-IF i= 0, 1h z + hznl p =1= 0 and IAI =j:. o. Assume J at j5 has a geometricOlly simple 1 eigenvalue with right eigenvector U and left eigenvector ut and there is no other ei:f!envalue 'With modulus 1. Then a fold bifurcation /1, 4, 6J occurs for (x n , Yn, Zn) of eqns.. (1)-(9) if at p
ut [
1)yq).. ] (U · U) D~~
where D2 9 rey
.[8
=
ut [ ~ ] :;:6 0,
~] 8x n 8Yn
2gn 2
8x n
~ 8Yn 8x n
+ h )-lh 0
g Zn ( h Z
¥- 0 and ~ 8Yn
&
fT" !!.f.B..
and 8t/J.,. = JO rr eJt 8a
Zn
~
·Ba --"Eh:
=
9
Q
dt~dt 8Ot:
•
Theorem 5. 2 Suppose that p == (x, x, y, z, Z, 0:) is a strict equilibrium of eq:ns.(1)-(3) and 1hz Ip # 0, 1hz + h zn Ii' t= 0 and IAI #- o. Assume J at 15 has a geometrically simple -1 eigenvalue with right eigenvector U and left eigenvector ut and there is no other eigenvalue with moduli 1. Then a flip bif4.rcation [1, 4, 6} occurs for (x n , Yn, Zn) of eqns.(1)-(9) if at p
[ut [ 2
D~y
D~lIgR
2Ut 8D xy cPr U
8a
] (U. U)]2 +
+ ut [
ut [ 6
D:x;y4J-r ]
DxygR
D;yrP1'" ] (U.
D xy 9R
ut [ ~ ]
Theorem. 5. 3 Suppose that 15 =
~
u . U) ¥
(U· U)
#
0,
Before presenting the singularity-2 induced bifurcapoint p. Assumption 5. 1 (Coordinate Condition) Assume . that there exist a column {3z in ma-
trix
(h ~z column in matrix 1 such that after replaces into z1\z.. ), and
( hfE
( h
h
z
",
1hz Jp i=
other eigenvalue with modulus 1. ...L 0 and ejk~(ii) -L 1 Jl.for k == 1, 2, 3, 4 , then ~ ~ there exists the Neimark-Sacker bifurcation which generates an unique closed invariant curve fn?m the strict equilibrium, for (xn , Yn, Zn) of eqns. (1)-(3). Theorems 5. 1, 5. 2 and 5. 3 are derived from eqns.(13)-(14) and the definitions of Fold, Flip and
Neimark-Sacker bifurcations. surface
is {(x n , X n , Yn, Zn, Zn, a) 11hz (x n , X n , Yn, Zn, Zn, 0:) I = O)}. At a strict equilibrium 15, from eqns.(1)-(3), we have f(x, x, y, z, z, a) == 0 (15) g(x, y, z, a) - fj == 0 (16) hex, x, y, z, Z, 0.) = o. (17)
j3fE
and a
)
z
fE h"'n )
j3z
{3",
hz' +hz:,. are nonsingular at
hz'
a point p for the alternated matrix ( h
where (h ' ~Jhz:, )
z
•
~Jhz:..
is constructed by n z
z
), -
1
~zhz", ) except the column {3z, z and by the column f3x. Then we describe the singularity-2 induced bifurcation at sampling instants. Theorem 5. 4 Suppose that 15 = (x, x, y, z, z, 0) is a strict equilibrium of eqns.(l)-(3) with onedimensional parameter a E R and the Coordinate Condition (Assumption 5.. 1) holds at 15. Assume that 1h z 115 has an algebraically simple zero eigenvalue and bz = -trace [ fzadj(hz)h x J is nonzero, Ilbzll> Ilbznll where bzn == -trace [ fznadj(hz)h:t: }, and furthermore the following conditions are satiscolumns of ( h
fied,
I
f~ + !"!X:n. 9';Cn
hx
+ hx:n
Ix +/z'n 9a:n
11 dr{a) do
singular
)
tion, we define a coordinate condition for h = 0 at
o.
0, Ih ot + h zn I~ =1= 0 and JAI -# 0.. Assume that J at P has an algebraically simple pair of eigenvalues in the form of ..\ = r(o - Q:)€±j~(a-a) where r(a) = 1 for all sufficiently small IQ - ai, and there is no
A
+ fz"",,
fz
eqns.(15)-(17).
(x, x, y, z, z, 0) is
a strict equilibrium of eqns. (1)-(3) and
fytt.
(18) gXn gtJn - E gZn ( h z + h Xn h Yn h z + h zn p for (x, y, z) of eqns.(15)-(17) is nonsingular. Then there exists an unique equilibrium locus (x(a), y(a), zen), et) E nn:2:+ n J,l+n.-:+l locally near 0: = 0, according to implicit function theorem and
h:z;
1
Cz
+ h- xn +
6 x n,
Iv'n
Iz
ht/on
h~
9vn - E
f:::
fYn
E
9Yn -
+ fz n + h%n
9zn
hz
6 11n
-Sz
lOll
+!;l!.n gz.n.
h Un
+ hzn. + Cz.n,
I =F 0, and
:;e o.
9 h_ ba
Then there. exists a smooth curve of equuibria in nn~+n~+n.=+l which passes through p and is transversal to the sin9vlar surface at p. When Q increases through et, one eiyenvalue of the system for (Xn , Yn, Zn) of eqns.(1)-(9) (i.e., an eigenvalu.e of J evaluated along the equilibrium locus) moves from less than modulus 1 to more than modulus 1 if bz/c > 0 (respectively, from more than modulu.s 1 to less than modulus 1 if bz/c < 0) by diverging through ±oo. The othe.r (n x + n y - 1) eigenvalues remain bounded and change in the order of O(Ct - a), where c == 60: - (8x +8xn , 8yn , Oz + bz
n
) (
J-r. + ha;
Ixn.
9:r:n + ha:n
ated at p.
fYn 9Yn - E h. Yn
iz h%
+ f%n
9z n
+
h,zn
)
-1 (
101. ) 90: ho.
evalu-
We call the bifurcations depicted in Theorem 5.4 as singularjty-2 induced bifurcations, in distinction from the singularity-l induced bifurcations de-
scribed in Theorem
5~
5.
The singularity-2 in-
duced bifurcations occur at sampling instants for
2393
Copyright 1999 IFAC
ISBN: 0 08 043248 4
NONLINEAR ANALYSIS OF SAMPLED-DATA CONTROL SYSTEMS...
14th World Congress ofIFAC
ancl v nI --
I I} x2,n , ~ 8v:a -1 ···,xnx ,n
gz~ (h Zl
+
(X n , Yn, zn). Before proving Theorem 5.4, we present the following lemma[5, 6]~ Lemma 5. 1 Assume the same conditions as The-
hz~)-l(hx,+hv~), ~ = 9Yn -gz~(hzl+hz~)-lhYn4
(x', zJ) for (x, z) with (x~, z~) for X n , Zn such that the matrices A, B, C for eqns.(7)-(9) are diffeomorphically transformed into the form A' , B',Cl respectively, along the equilibrium locus at the neighbourhood of15
Therefore, when a: increases through 6:, one eigenvalue of the system (i.e~, an eigenvalue of J evaluated along the equilibrium locus) moves from less
{
gv~
-
* is an appropriate matrix with each element having orem 5. 4, then there exists a local coordinates the form of constant + O(Q - a)~
A'
(~+o(a:-a)
-
c\a-~J
-
B'
c(~~~a) + CJ(o: - a) C + 0(0 - a)
=:: (
0
CZn,
Zn
C1
0
_(
-
+ O(n -
C Y1 ,n
than modulus 1 if bzlc < 0) by diverging through ±oo. The other (n x + n y - 1) eigenvalues remain bOWlded, change in the order of O(a - a) and stay close to those of Cl due to bz i= 0, c ~ 0 and
0 ) ex +O(a-a) ,
0
C1J~
a)
)
+ 0(0 -
a)
IlbzU > Hbznll.
,
0a))
+ 0(0: -
where x' == x'(x,xn,Yn,z,Zn,Q) E Rnz,z' z'(x,xn,Yn,z,Zn,ll') E Rnz are Coo mappings
in the open neighbourhoods of jJ and C x is a nonsingular matrix. y~ = {Y2,n, ... , Yn~,n}. ex, CZn.' CXn ' CY1,n. and Cy~ are (n x - 1) x (n x 1), (n;z: -1) x 1, (n x -1) x (n x -1), (n x -1) x 1 and (n x - 1) x (n y - 1) matrices respectively, which are all independent of (a - a). Proof of Theorem 5.4. ''''le first show that the equilibrium locus is transversal to the singular surface. Since the matrix of eqn.(18) is nODsingular, there exists an unique equilibrium locus (x(a), y(a), z(a:), et) E R.n~+nll+nz+l locally near Q = Q. Then along the equilibrium locus at a, c~
Jz
+ f~n +
9$n. h~n
i'lln
9Yn h Yn
f%
E
h z
+ fZ n
+
e (
c(:~Q) + 0(0 o
go. ha
hzt'),
0
Ci) e TCz
)
+ CJ(a: -
-
cQc.;lc.t::n
c O C; l
(
1)
+
C '1tl •
+ 0(01
0(0 -
~+
-
a)
0
coC.;l CZTJ
_
O(CIl _ a)
COC;lC 1
~ lJv n
Example 5.1
(nr
+ 0(0
-
a}
where
)
tin
0(0 -
&)
..
COC~lCx.,...
Wlstable manifold and one dimension invariant sin-
0
b~t:%D" c(:~"') _- ~ +
_ ( Cl -
It can be shown from Theorem 5~ 4 that the dynalnics can be locally decomposed into nB-dimension invariant stable manifold, a nU-dimension invariant
0(0 ~ 0)
respectively, due to Lemma 5. 1, where Co = E. Hence at the transformed coordinates (x',x~,Yn,z',z~), the Jacobian matrix J for (x~, Yn) has the form J',
where
while the other (n x + n y · - 1) eigenvalues remain bounded and change in the order ofO(et - et).
)
+
eTC:z; -
= (
• from less than -1 (or -00) to less than modulus 1 i.f bz+b zn < 0 and ~ < O· 'J bz; C 1
Yn+l
a)
a}
I
c'
gular manifold near p where n S and nU be the numbers of stable and unstable bounded eigenvalues of J[5, 6].
according to Lemma 5.1, and furthermore (erA E)A- 1 B and (e'TA - E)A-1C have the forms
~ (e c(::l'a)
• from more than 1 (or 00) to less than modulus 1 zf bz+b zn > 0 and ~ < o·
d erlVe ~ cl
fa)
) -1 (
Slzn.
transformed into
JI
• from less than mod'Ulus 1 to less than -1 by diverging through - 0 0 if b~t~%n < 0 & ~ > 0;
bz
gt!lar surface 1h z I = O. Without loss of generality, next we take the local coordinates (x', z ) with lX~, z~) of Lemma 5~ 1 at p. Then at the new local coordinates, eTA can be
a.nd
• from less than modulus 1 to morn than 1 by diverging through 00 if b:r:t~%n > 0 & ~ > 0;
(19) (20)
0
from eqns.(15)-(17) and eqn4(18) is substituted into eqn.(19). In eqn.(20), Schur's Theorem is also used from the assumptions. Therefore, eqn.. (20) means that the equilibrium locus is transversal to the sin-
(
Remark 5. 1 Assume the same conditions of Theorem 5.4- . When a increases through 0:, one eigenvalue of J evaluated along the equilibrium locus moves
(~,~. ~)T
\vhere h~
~, ~~)T
Zn ) ( ; : ,
#
(
0
Assuming the same conditions as Theorem 5.4 but with IJb z 11 ::; Ilbz 11, it is evident that the singularity-2 induced bir"u.:cations still occur but are degenerate according the proof of Theorem 5. 4 . According to Theorem 5.4 and the Jacobian J' of eqn.(21), we have the following rernark~
£;-6(x(Q), x(o:), yea), zea), z(Q), et)
+ (5x + oZn' 6yn , 5z + C
00.
-
>0 (respectively, from more than modulus 1 to less
than modulus 1 to more than modulus 1 if bz/c
COC;-101l1 .,...
Cl
+
)
o~o: ~ a)
COC;;lC1l £S.B. n 8Yn
) ,
Z~
Zn
~
x
=
= 0'2 - Z + Yn - a O~5Yn + X n - 0:4 + 0.50:
(22) (23)
x - z2 = 0 t < (n + 1)1"; n == 0, 1, 2, .~.)
Lt is a bifurcation parameter, and x, and a E 'R 1 •
(24) Xn ,
Yn,
Obviously x = xn = 0 4 , Yn == a: and Z == zn = 0'2 are a strict equilibrium of eqns.(22)-(24) for any a E 'RI. Let a = 0 and p = (0,0,0,0,0,0), (x(a),y(a),z(a),a) = (a 4 ,a,a 2 ,a) is an wlique (21~quilibliwn locus in n 4 locally near et = O. It is easy to check that eqns~(22)-(24) at p satisfies the conditions of Theorem 5. 4 which means that the singularity-2 induced bifurcation occurs at p when et changes.
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5.2
Intersampling instants
In this section, we will show that all of the bifurcations except a "degenerate" singularity induced bifurcation for a strict equilibrium at the intersampling instants are related to those at the sampling .instants. In other words, there is no independently local bifurcations at the intersampling instants, except the singularity induced bifurcations which include both independent and dependent bifurcations on those of the sampling instants. The stability of the DDA is determined by J which is the Jacobian of the equivalent discrete system for eqns.(1)-(3) according to Theorem 3. 2 ~ Therefore, there is no biflll'cation at the intersampling instants as far as 1h z I =f. 0 at p. Although the stability of eqns.(1)-(3) is reined by J, the trajectory of (xCt), z(t)) at the intersam-
pling instants follows eqns.(l) and (3) which is a differential-algebraic model.
Theorem 5.5 Suppose that 15 = (x,x,y,z,z,a) is a strict equilibrium of eqns,,(1)-(3) with onedimensional paramete.r 0: E R~ Assume that lhzl p has an algebmically simple zero eigenvalue and bz = -trace { fzadj(hz)h x } is nonzero, there is no purely imaginary eigenvalues for Jacobian matrix A and furthermore the following conditions are satisfied, Ix fz fa and h x h z ha: #- o.
{: I~o,
6x
8z
Ca
Then there exists a smooth curve of equilibria (of eqns.(l) and (3)) in R n :r+n z+ 1 which passes through 15 and is tmnsversal to the singular surface at jJ. When Q increases through a, one eigenvalue of the system for (x,z) ofeqns~(l)-(3) (i.e., an eigenvalue of A evaluated along the equilibrium locus) moves from c- to C+ if bz/c > 0 (respectively, from C+ to if bz/c < 0) along the real axis by diverging through oo~ The other (n x - 1) eigenvalues remain bounded and change in the order of 0(0'-0), where
c-
C
= Oa - (oz, 6z )
(~:
{:) -1
(
{: )
evaluatOO
at p. We call the bifurcations depicted in Theorem 5. 5 as singularity-l induced bifurcations which occur at intersampling instants. Remark 5. 2 All of the local bifurcations except singularity-induced bifurcations at the intersampling instants are related with those of the sampling in.sta·nts.
5.3
Genericity of local bifurcations
This section will show that the DDA, eqns. (1)-(3) has five generic bifurcations described in Theorems 5. 1 -5. 5 under certain generic assumptions. We first give several definitions for structurally stability and bifurcation set [1]. Definition 5. 1 A diffeomorphism map or vector field F(x) : X ---+ X E n m is called structurally stable in X if there is an t: > 0 such that whenever sUPzEx{IIF(x) - G(x)11 + IIFz(x) - Gx(x)11 < E} for G : X ---+ X) F is topologically equivalent (conjugate) to G] in other words, there exists homeomorphism H = X --+ X such that Ho F == Go H. De:finition 5. 2 The bifurcation set is the set of po~nts for which the dynamics is structumlly unstable under variation of pammeters~
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Let EP = {x n , Yn, Zn, et} and BP = {x n , yn.~ i~, a} denote the set of strict equilibria and the bifur-
cation set of the strict equilibria for eqns~(1)-(3),
respectively. Therefore EP = {(x n , Yn, Zn, a) : f(p) == 0, Yn == g(p), h(P) = 0 where p == (x n , X n , Un, Zn, Zn, o:)} which is clearly an open subset of nn:c+ny+nz:+l. Define the Hurwitz matrix H rn - 1 as al
a3
a2
a5 a4
a2m.-3
ao 0 0
al
aa
a2m-5
a2
a4
a2m.-6
o
0
0
am-l
a21n-4
(25)
and H rn - 1 (AE-A) means that the elements
ai
cor-
respond to the efficients of polynomial lAE - AI == aoA7'n+ a1 .A. Tn - 1 + ~.~ + am. for m x m matrix A, while Hm-l(~E-A) means that the elements ai correspond to the efficients of I ~=~ E - AI == (ao..\Tn + atAtn-1 + ... + am.)!(l - A)1n-l for m x m matrix A. IH~-l(AE - A)I ~ 0 implies that A has purely imaginaryeigenvalueswhile IHm-l(~!~E-A)i= 0 means that A has modulus 1 eigenvalues. Then we directly have the following lemma. Lemma 5. 2 The bifurcation points for strict equilibria of eqns~(1)-(3) are composed by four sets BP = BP n (SSI U 8PD U SFP U SNS) (26) where SSI = 8FD
= O}
{(:r:n.. Yn~.a:n, Q) E EP : Ihzl {(:Xn. 1.'n., Zn. 0) E EP : IJ - EJ {(~nl Un. %nt 0:) E BP: IJ El
=
8FP
=
SNS
=
+
{(:Z:ndln; zn.O:) E EP:
=
D}
= O}
IHna:+nJil-l(~E- J)I =
(27) (28)
(29) O}.(30)
Notice that the system does not generally undergo bifurcations at the set {(x n , Yn, Zn, a) E EP : 1hz + h Zn I = O}, because one of the bifurcations in Theorems 5. 1- 5.4 will occur before 1h z + h zn I
approaches zero~ It is evident that each set in Lemma 5. 2 is restricted by only one equality on EP~ Let Ssil, Std, Sip, Sns and Ssi2 be the sets of points satisfying the conditions of 5. 5 ,5. 1 ,5. 2 ,5. 3 and 5. 4, respectively. Then, we suppose the following generic assumptions. Assumption 5. 2 Ssil, S Id, Sun Sns and Ssi2 are open and dense in SSI, SFD, ~FP, SNS and SSI, respectively~
The generic bifurcations of eqns.(l)-(9) are constituted by the bifurcations of Theorems 5. 5 ,5~ 1 ,5. 2 ,5~ 9 and 5~ 4 under Assumption 5.2. Proof of TheorelIl 5. 6 . Since only additional independent non-equalities are restricted on 8 s i1, Bid, SIp, SnB frOIIl 8S1, SFD, SFP, SNS, due to Assumption 5. 2 we get Theorem 5. 6
Bid = SPD,S/p:=::: SFp,Sns ~ SNS,S8il = SSI (31) where S means the closure of a set S. Moreover compared with Ssil, Ssi2 is also constrained by an inequality Jlbzfl > IJbzn. II in addition to several nonequalities from SSI. Hence 8 si2 C Ssil. Considering that Ssi2 is also a dense and open subset of SSI according to Assumption 5.2, we express BSI by B sil U Ssi2 = BSI. Therefore according to Lemma 5. 2 , we have BP == BP n (Bsil U Ssi2 U Bid U SJp U Sns) (32)
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theorern~ 0 Singularity-l induced bifurcation can generally be
which proves this
expected to occur when the system undergoes the singuiarity-2 induced bifurcation due to Ssi.2 C Bsil , although they are restricted by different inequalities and non-equality, Le~ Ssi2 is not a subset of Sail. However, most of the independently locally bifurcations at the intersampling instants are composed of the degenerate singularity-l induced bifurcations as shown in the proof of Theorem 5. 5 , according to the genericity from Theorem 5. 6 . Remark 5. 3 Most of locally nondegenerate bifurcations for the DDA at strict equilibria are related with the local bifurcations of the sampling instants
under Assumption 5.2.
6
Application to Power Systems
We show how a power system with digital PSS can be treated as a DDA in this section. For a power system, the dynamics of the generators and analog controllers(e~g" AVR) is defined as eqn.(l) of the DDA, while eqn.(3) is composed of the power bal-
ance equations. Then eqn.(2) corresponds to the discrete-time controllers (or digital PSS). In this paper, the parameters of the digital PSS are first designed by imitating the conventional analog PSS, and then are adjusted and evaluated by using Theorems 3.. 2 -5. 4 . Next in order to obtain the digital PSS by imitating the designing of analog PSS, we derive eqn. (2) from an analog PSS with a hold ciruit and a sampler. Generally, this process is not necessary if the digital PSS or eqn.(2) is given. Generally, the digital PSS consists of a sampler, a discrete-time controller and a hold circuit. In this paper, we adopt three-order digital PSS with a zero-. order hold circuit which is obtained by discretizing the analog PSS~ Le. "Control device". The state y(t) E 1(,3 and input Un E 1l of the PSS follow D1y(t) + D2Un U.(Xn , zn) == u(x(nT), z(nr)) Un == (nT ~ t < (n + l)T; n == 0, 1, ...) (33) and the output V n E 'R of the digital PSS can be expressed as
yet) =
v(t) =
-
{
Vma.:t
if v(t) ~
Vma:r
Vm.in
If v(t) $
Vnt.in
~:~: VI (t)
(nT :S t
+ Ry~(t) + 1/3(t) + ~~~ Un
< (n + 1)7; n
Vn
14th World Congress of IFAC
Yn+l == eD~TYn + D 1 1 [e Dl 7" - E]D2u(xn , zn) (35) which has the same form with eqn,,(2). That is, eqn~(35) is the discrete-time controller whose design imitates that of the continuous-time controller, i.e~ a three-order analog PSS. Note that the real power output u(x n , zn) of the generator is a given function of (x n , zn). The PSS output V n is obviously a f\ll1ction of '})n and u(xn , zn) according to eqn.(34), Le., Vn
7
[
_*';2~~a -
T4 -zT5
2
T4
-
_
4
D2
= [
conditions~
Finally, the theoretical results were ap-
plied to the evaluations of digital control for power systems.
References
}
4t ¥t[a ]. KT;
Conclusion
the system unstable, and derived their bifurcation
are upper and lower
>
(36)
then, this paper showed that there are five types of
== VenT)
* :]'*
T4'1-' T p 4
=== 71(Yn, u(x n , zn))
generic bifurcations at strict equilibria which cause
bOlillds for the internal output of the PSS. Un and V n are the values of u(t) and v(t) at instant t = nT respectively, which are constantly hold during nr ~ t < (n + 1)r. Define D 1 and D 2 as follows
D1=
Zn)
We first described locally sampling, asymptotical stability anq. singular perturbation for DDA. And
= o~ 1, ...),
V'Jna~,.VJl1,in
7Jn (x n , Yn,
where V n is not a linear function because of the saturation Vmin,Vrnax of vet) shown in eqn.(34). On the other hand, the dynamics of generators and analog controllers corresponding to eqn.(l), the power balance equations corresponding to eqn.(3) can be written as follows x(t) = f(x(t), 17n(Xn , Yn, Zn), z(t)) (37) h(x(t), l1n(Xn )Yn) Zn), z(t)) == O~ (38) Notice that X n , Yn and Zn can be viewed as constants during nT :5 t < (n + 1)7 for eqns~(37)-(38) because Un is constantly hold during that period. Eqns.(37)-(38) are generally nonlinear. Therefore, combining eqns~(37)(35)(38),we can use the Theorems 3.2 -5.4 to analyse the power systems with the digital PSS nwnerically in a nonlinear way. Several numerical simulations have verified that our results are useful[S, 6].
(34) where, yet) = (Yl(t)'Y2(t)'Y3(t))T . u E n is the real power output of the generator which is also a given function of (x, z)~ v E 1?, (or vn E 'R) is the PSS output, and
=
T 2_ Ti
~~ 4
Since eqn~(33) is a linear differential equation, it can be analytically solved. Therefore, we have following difference equation by integrating eqn~(34) from fiT to (n + 1)T,
[1] J. G uckenheimer , P"Holmes.(1983). Nonlinear Oscillations, Dynamical Systems, and Bifurca-
tions of Vector
Fields~
Springer- Verlag, New
York.
[2) V.Venkatasubramanian, H.Schattler, J.Zaborszky~ (1992)~ A stability theory for large differential algebraic systems. Report SSM 9201-Part 17 Department of Systems Science and Mathematics, Washington University-, St.Louis,MO. [3] V.Venkatasubramanian, H.Schattler" J.Zaborszky~ (1995). Local Bifurcations and Feasibility Regions in DifferentialAlgebraic Systems. IEEE 'Prans. on AG, 40, No.12, pp~1992-2013. [4] L.Chen, K.Aihara. (1997). Chaos and asymptotical stability in discrete-time neural networks. Physica D, 104, pp.286-325. [5] L.Chen, K.Aihara. (1997). Local Stability in Hybrid Dynamical Systems. 1997 International Symposium on Nonlinear Theory and its Applications (NOLTA '97), pp.73-76. [6] L.Chen, K.Aihara. (1998). Stability and Bifurcation Analysis of Hybrid Dynamical Systems. Technical Reports, METR 98-02, The University of Tokyo. [7] R~L.Grossman, A"Nerode, A.P.Ravn, H.. Rishel (edited). (1993). Hybrid Systems. SpringerVerlag, New York..
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E-2c-13-4
Copyright © 1999 IF AC 14th Triennial World Congress, Beijing, P.R. China
Nonlinear Analysis of Sampled-data Sy~tems a
a
Co~trol
bKazuyuki Aihara
Luonan Chen
Osaka Sangyo University,3-1-1 Nakagaito, Daito, Osaka 574-8530,Japan.
Fax:81-798-70-8189;E-mail:[email protected] bThe University of Tokyo., 7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656,Japan
Abstract: This paper treats a digital-controlled Control system with both continuous-time and discretetime variables as a model of differential- difference-algebraic equations (DDA)~ presents a fWldamental analysing method of such a nonlinear DDA for local sampling, asymptotical stability, singular perturbations and bifurcations, and further shows that there exist five types of generic bifurcations at the equilibria in contrast to two types in continuous-time dynamical systems and three types in discrete-time
dynamical systems. Finally the theoretical results are applied to digital control of power systems. Numerical simulations have verified that our results are usefuL Copyright© 1999IFAC
Keywords: asymptotic stability, singular perturbations, sampled-data control, nonlineax analysis, nonlineax theory, power s~stem, bifurcation
1
Definition of DDA
2
Introduction
Many physical systems with digital devices require
mathematical descriptions not only with ordinary
The following notation is used throughout the pa-
per. Let
ference equations. As an example, for the power system with digital controllers, the dynamics of the generators as well as their continuous-time
controllers and the load dynamics together define the ordinary differential equations while algebraic
equalities are defined by the power balance equations of the transmission network[3]. On the other hand, the difference equations describe the dynamics of the digital devices. Therefore., the digital control systems of power grids are typical models mixed with both continuous and discrete time sequences which can mathematically be formulated as differential- difference-algebraic equations (DDA). In this paper, we define the constrained sampled-
data system as a model of DDA. So far, both continuous-time and discrete-time nonlinear systems have attracted considerable attention and a variety of techniques have been developed[3, 7, 5]. This paper aims at analysing the asymptotical sta-
bility and bifurcations of the DDA (5, 6] and further applying the theoretical results to digital control of
power systems. We first define a DDA model and two types of equilibria, namely quasi-equilibria and
strict equilibria due to its mixing nature of continuous and discrete time, and summarize the recent
results for locally sampling, asymptotical stability and singular perturbations of DDA[5, 6]. Then this paper shows that there exist five types of generic bifurcations at the strict equilibria in contrast to t,vo types at an equilibrium in continuous-time dynamical systems and three types at a fixed point in
discrete-time dynamical systems, and provides their bifurcation conditions for the DDA. Furthermore, this paper indicates that most of locally nondegenerate bifurcations for a DDA at strict equilibria are related with the local bifurcations of the sampling instants under some generic assumptions. Finally, the theoretical results are applied to the evaluations and designs of digital control for power systemsA
= f(x(t), x n , Yn, z(t), Zn, 0)
(1)
a) h(x(t),xn,Yn,z(t),zn,a) = 0
(2) (3)
x(t)
differential-algebraic equations, but also with dif-
Yn+l = g(x n , Yn'
Zn,
(nr:5 t < (n + l)r;n = 0,1,2, ...) be the differential-difFerence-algebraic equations (DDA) under study, where vectors x, X n , Yn,
z
and
Zn
belong
to
R,n z , Rnf,t; , Rn lJ , Rn;& and
the
Euclidean spaces
nn~, ~pectivelYA T
is a
sampling interval (non-negative real number), and 0: E R is a parameter. Define X n = x(nT),Zn ==' z(nr) and Yn is the value at the instant ni, where X n , Un and Zn are constantly hold during nT :5 t < (n + l)r for eqns4(1) and (3). Then eqns.(1)-(3) are an ordinary differential - difference - algebraic equation model (DDA). Let !:r. = af(x~x8:1'l,Z1zn) and
1/-:£1
is the determinant of /z.
11·11
is the usual
Euclidean norm of a vector or the induced matrix nonn of a matrix. E is an identity matrix. eA is the exponential fllllction of a matrix A. In this paper, a non-equality is an equation with the form of u # 0 while an inequality is the form of u < 0 or u > O. On the· other hand, an equality is an equation with the form of'U == O. x{t) is continuous due to eqn.(l) while z(t) is generally discontinuous at t == nT, n = 0, 1 1 2, .. +' namely limt_nT-O x(t) = x(nT) == X n = limt---+nr+O x(t) and limt-nr-o z(t) #- z(nT) == Zn = limt---+n.+o z(t), as far as 1hz 1 =I- 0 and 1hz + hz~J i- O. In this paper, sampling instants mean t == nT while intersampling instants imply nT < t < (n + l)T~ where n == 0,1,2, Definition 2.1 Vectors p == (x,x,Yn~ z, z) are called a strict equilibrium of DDA eqns. (1 )-(3) if /(x,xn,Yn)z,zn,a;) == 0, h(x,xn,Yn,z,zn,O:) == 0 .'Aa
and
Yn == g(xn,Yn,zn,a).
Next, without confusion, t is occasionally dropped from the related variables, e.g., xCi) or z(t) is simply expressed by x or z +
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Local Sampling and Stability
3
Let p == (x n, Xn,Yn, zn' Zn) satisfy eqn,,(3) and Jhz + h Zn Ij =1= O~ Then, ~cord.ing ~o the implicit function theorem there eXIsts an unique Coo mapping in open neighborhoods of p such that at sampling instants (4)
C ~
-
p
z(t) = q(x(t), X n , Yn) nT ~ t < (n + l}T (5) where qn(xn , Yn) == q(x(nT), Xn , Yn) due to Zn == limt--+n7"+O z(t). Thus the reduced. system of eqns.(l) and (3) at the neighbour of p becomes
== I(x, X n , Yn, q(:z:, X n , y.",), qn(xn~ Un))
Yn+l == 9R(X n , Yn)
=g{x n , Yn,
qn(X n , Yn))
1hz I #
0, 1h z
(9)
+ h zn I:/;O at p.
strict-equilibrium of DDA eqns. (1)-(3)"
and further the reduced system of eqns.(2) and (3) at the neighbour of p becomes forn=O,l,~u
-
where GX = 9 X n - gz.n (h z + h ZtlJ- 1 (h x + h:c n ) and GY == 9Yn - 9zn.(h% + hZn)-lhyn.. ~en we h~ve the following theorem for the stabihty of a stnct equilibrium p for all of instants. Theorem. 8.2 Assume that p= (x,x,f),z,z) is a strict equilibrium of DDA eqns. (1)-(9), and 1hz Jp =f; 0, 1hz + h Zn 1p =I- 0, lA) # O. 1. If all the eigenvalues of J at 15 have moduli less than 1 J then p is an asymptotically stable
such that at the intersampling in-
stants
x :::: fR(X t xn,Yn)
(8)
_
hz..,.(h% + hzn)-lhvn ] JZn (h z + h Zn )-lhvn
Assume IAI =F 0 and Define a J acobian
Furthermore, let p satisfy eqn.(3) and Ihzf.,; "# o. Then by using the implicit function theorem again, there exists an unique COCJ mapping in open neighborhoods of
-IZn{hz;+h:",)-l(hx+h zti )
Ivn. - Jz h;-l[hYn
2~
(6)
If anyone of the eigenvalues of J at p has modulus more than 1, then p is an unstable strictequilibrium of DDA eqns.(1)-(9).
Actuallr, we can prove that Theorem 3. 2 still holds even if IAt = o.
Let ~&, ~8 ~ and ~ are partial derivatives Z X n UYn of !R(X,Xn,Yn) at (x,Xn,Yn) with respect to (x, X n , Yn), respectively~ Then we obtain a locally sampling theorem [5, 6]. Theorem 3. 1 Assume that p = (x, x, y, z, z) is an arbitrary point which satisfies eqn~ (8). Assume that the partial derivatives of f unto appropriate
4
Singular Perturbation
For the DDA eqns.(1)-(3), we define an associated singularly perturbed system aB x(t) =
f(x(t) , X n , Yn, z(t), zn)
Yn+l =
g(xn,Yn,zn)
(10) (11) (12)
order exist, and 1hz I # 0, 1h z + h zn ~ #- 0 at any €z(t) = h(x(t), X n , Yn, z(t), Zn) solution of eqns~(l) and (3) starting from {J. Then (nr ~ t < (n + l)-r; n = 0, 1, ...) the discrete-time system for eqns. (1) and (8) can be where € is a sufficiently small positive number. That e.xpanded in a neighbourhood oip for (Xn,Yn), is, the algebraic eqn~(3) corresponds to a limit of the 1 2 (~) X n +l ~ rP7+Dxy tPT ~XY+2Dxy.p'T' Axy·~xy+... n ~ 0,1, ... fast dynami~ eqn.(12). Obviously, eqn.(12) will ap-
=
where" 4!.r:.
fT(x, Yl
x + J: iRdt
=
fR(rPt,X, YJ an rPo = X. ~%!I = [ : : : ~ ] . Dxy40T = [~. 2
[~2;c\T ~
_
D~'YtPT-
tin
~ = 8x-n. r
T
Jo
a~: tin ] ~
JR
proach the algebraic manifold when
.
positive)" Furthermore, assume that no eigenvalue of J at p has the modvlus 1, and 1h z + h Zn I" i:-
eJo'" ~dt + Jor eJ~'" *"dt!!lE.dt, 8xn. T
[}X
T
8x
0, IAl p # Ow Then there exists f > 0 such that for all positive £ < ~, stability of eqns. (10)- (12) at p is identical to that of eqns~ (1)-(9).
82~T
rT eft ~dt[82l,(!!.!l!t.-)2+
Jo
8x""
n
+~~+ 8x8z'f'I, 8z a
~dt(a2ir!!ft!!i!J.. + 8x 8y". 8x ~~+~E!&+~)dt ~ = fJx8Yn 8x n 8z8xn 8yn. 8x n 8Yn ' 8Ynoxn Ox""
~ ==
2
8 t""R 8
'
8x n 8Yn
eh
T
fr Jo
8x8xn. 8Yn
+
2
2
8 fR f!..P..t. +_El jR)dt
tJYn8x OX n
~,
8 2
_
-
- Neimark-Sacker and singularity-2 induced bifurcations at the sampling instants, in codimension one
fT '!JJ1. at 2 ~ 2 ~ 2 ~ 2J for eJt 8X (~(U::)2+~~+~M;:;+~)dt. Jo 8a; tlfl ~ Un lIn Un x '!In 811n.
Theorem 3. 1· ensures that the discretized system can locally approximate the original continuous system "at any order at sampling instants. At a strict equilibrium p, if h z and h z + h zn are nonsingular A B
==
=
Ix
-Izh~lhx
J~n - fzh;l{h xn - h%n (h.
parametric space.
Zn
Sampling instants
5.1
For the sampling instants, eqns.(1)-(3) at the neigh= (:15, X, y, z, z, et) can be expressed as bourhood of a strict equilibrium p
(7)
+h
Local Bifurcations
We will show that there are five types of bifurcations for a strict equilibrilllll tlllder certain generic conditions, Le., singularity-l induced bifurcations at the intersampling instants, and fold, flip,
+
ex; &ij;;
8%
+~~
5
n
r-r eft'T ~dt(82jf~~
J0
O. It is ev-
Theorem 4. 1 Assume that 15 = (x, x, y, z, z) are a strict equilibrium of the DDA eqns. (1)-(3), and all real parts 0/ eigenvalues for hztp are negative (or
81J n
:::tn.
E ~
ident that eqns.(10)-(12) has the same equilibrium as eqns.(1)-(3) as far as h = O.
==
~l,
eh'" *t-dt'lla ay:;; dt )
,p~R]dt
with
)-lhzn ]
.:t:'n+l "n.+l
= =
,pT
+
Dxya~7" A xll Ot.
+
~D~yo4',.- L:.;rya: - .a.:rg~ (n=D.l •... ).
9(:;Cn.Yn~qn(=nIYn.Q),Q)
+ .. ,
(13) (14)
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T
where
cPr =
!R(X, x,
y, a)
+
If
J; fR(cPt,x,y,a)dt
and ~zya:
=
Xn [
x+
==
-x ]
Yn - ~
,when
Q-O:'
Jh z Jp i= 0 and Jh z + hzn)p I- O. Since a strict equilibrium of the DDA eqns.(1)-(3) is a fixed point of discrete time system eqns.(13)(14), the bifurcations of eqns.. (1)-(3) at the sampling instants for the strict equilibria are basically
14th World Congress ofIFAC
Suppose that the full Jacobian matrix Ix + fX n
the same as local bifurcations of the differencealgebraic model eqns.(13)-(14). Next we examine the fixed point bifurcations of eqns.. (13)-(14). We first swnmarize the fold, flip and NeimarkSacker bifurcations [1], and then present the existence of singularity-2 induced bifurcation as well as their conditions.
Theorem 5. 1 Suppose that p = (x, x, y, z, z, a) is a strict equilibrium of eqns.(1)-(3) and 1h z-IF i= 0, 1h z + hznl p =1= 0 and IAI =j:. o. Assume J at j5 has a geometricOlly simple 1 eigenvalue with right eigenvector U and left eigenvector ut and there is no other ei:f!envalue 'With modulus 1. Then a fold bifurcation /1, 4, 6J occurs for (x n , Yn, Zn) of eqns.. (1)-(9) if at p
ut [
1)yq).. ] (U · U) D~~
where D2 9 rey
.[8
=
ut [ ~ ] :;:6 0,
~] 8x n 8Yn
2gn 2
8x n
~ 8Yn 8x n
+ h )-lh 0
g Zn ( h Z
¥- 0 and ~ 8Yn
&
fT" !!.f.B..
and 8t/J.,. = JO rr eJt 8a
Zn
~
·Ba --"Eh:
=
9
Q
dt~dt 8Ot:
•
Theorem 5. 2 Suppose that p == (x, x, y, z, Z, 0:) is a strict equilibrium of eq:ns.(1)-(3) and 1hz Ip # 0, 1hz + h zn Ii' t= 0 and IAI #- o. Assume J at 15 has a geometrically simple -1 eigenvalue with right eigenvector U and left eigenvector ut and there is no other eigenvalue with moduli 1. Then a flip bif4.rcation [1, 4, 6} occurs for (x n , Yn, Zn) of eqns.(1)-(9) if at p
[ut [ 2
D~y
D~lIgR
2Ut 8D xy cPr U
8a
] (U. U)]2 +
+ ut [
ut [ 6
D:x;y4J-r ]
DxygR
D;yrP1'" ] (U.
D xy 9R
ut [ ~ ]
Theorem. 5. 3 Suppose that 15 =
~
u . U) ¥
(U· U)
#
0,
Before presenting the singularity-2 induced bifurcapoint p. Assumption 5. 1 (Coordinate Condition) Assume . that there exist a column {3z in ma-
trix
(h ~z column in matrix 1 such that after replaces into z1\z.. ), and
( hfE
( h
h
z
",
1hz Jp i=
other eigenvalue with modulus 1. ...L 0 and ejk~(ii) -L 1 Jl.for k == 1, 2, 3, 4 , then ~ ~ there exists the Neimark-Sacker bifurcation which generates an unique closed invariant curve fn?m the strict equilibrium, for (xn , Yn, Zn) of eqns. (1)-(3). Theorems 5. 1, 5. 2 and 5. 3 are derived from eqns.(13)-(14) and the definitions of Fold, Flip and
Neimark-Sacker bifurcations. surface
is {(x n , X n , Yn, Zn, Zn, a) 11hz (x n , X n , Yn, Zn, Zn, 0:) I = O)}. At a strict equilibrium 15, from eqns.(1)-(3), we have f(x, x, y, z, z, a) == 0 (15) g(x, y, z, a) - fj == 0 (16) hex, x, y, z, Z, 0.) = o. (17)
j3fE
and a
)
z
fE h"'n )
j3z
{3",
hz' +hz:,. are nonsingular at
hz'
a point p for the alternated matrix ( h
where (h ' ~Jhz:, )
z
•
~Jhz:..
is constructed by n z
z
), -
1
~zhz", ) except the column {3z, z and by the column f3x. Then we describe the singularity-2 induced bifurcation at sampling instants. Theorem 5. 4 Suppose that 15 = (x, x, y, z, z, 0) is a strict equilibrium of eqns.(l)-(3) with onedimensional parameter a E R and the Coordinate Condition (Assumption 5.. 1) holds at 15. Assume that 1h z 115 has an algebraically simple zero eigenvalue and bz = -trace [ fzadj(hz)h x J is nonzero, Ilbzll> Ilbznll where bzn == -trace [ fznadj(hz)h:t: }, and furthermore the following conditions are satiscolumns of ( h
fied,
I
f~ + !"!X:n. 9';Cn
hx
+ hx:n
Ix +/z'n 9a:n
11 dr{a) do
singular
)
tion, we define a coordinate condition for h = 0 at
o.
0, Ih ot + h zn I~ =1= 0 and JAI -# 0.. Assume that J at P has an algebraically simple pair of eigenvalues in the form of ..\ = r(o - Q:)€±j~(a-a) where r(a) = 1 for all sufficiently small IQ - ai, and there is no
A
+ fz"",,
fz
eqns.(15)-(17).
(x, x, y, z, z, 0) is
a strict equilibrium of eqns. (1)-(3) and
fytt.
(18) gXn gtJn - E gZn ( h z + h Xn h Yn h z + h zn p for (x, y, z) of eqns.(15)-(17) is nonsingular. Then there exists an unique equilibrium locus (x(a), y(a), zen), et) E nn:2:+ n J,l+n.-:+l locally near 0: = 0, according to implicit function theorem and
h:z;
1
Cz
+ h- xn +
6 x n,
Iv'n
Iz
ht/on
h~
9vn - E
f:::
fYn
E
9Yn -
+ fz n + h%n
9zn
hz
6 11n
-Sz
lOll
+!;l!.n gz.n.
h Un
+ hzn. + Cz.n,
I =F 0, and
:;e o.
9 h_ ba
Then there. exists a smooth curve of equuibria in nn~+n~+n.=+l which passes through p and is transversal to the sin9vlar surface at p. When Q increases through et, one eiyenvalue of the system for (Xn , Yn, Zn) of eqns.(1)-(9) (i.e., an eigenvalu.e of J evaluated along the equilibrium locus) moves from less than modulus 1 to more than modulus 1 if bz/c > 0 (respectively, from more than modulu.s 1 to less than modulus 1 if bz/c < 0) by diverging through ±oo. The othe.r (n x + n y - 1) eigenvalues remain bounded and change in the order of O(Ct - a), where c == 60: - (8x +8xn , 8yn , Oz + bz
n
) (
J-r. + ha;
Ixn.
9:r:n + ha:n
ated at p.
fYn 9Yn - E h. Yn
iz h%
+ f%n
9z n
+
h,zn
)
-1 (
101. ) 90: ho.
evalu-
We call the bifurcations depicted in Theorem 5.4 as singularjty-2 induced bifurcations, in distinction from the singularity-l induced bifurcations de-
scribed in Theorem
5~
5.
The singularity-2 in-
duced bifurcations occur at sampling instants for
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NONLINEAR ANALYSIS OF SAMPLED-DATA CONTROL SYSTEMS...
14th World Congress ofIFAC
ancl v nI --
I I} x2,n , ~ 8v:a -1 ···,xnx ,n
gz~ (h Zl
+
(X n , Yn, zn). Before proving Theorem 5.4, we present the following lemma[5, 6]~ Lemma 5. 1 Assume the same conditions as The-
hz~)-l(hx,+hv~), ~ = 9Yn -gz~(hzl+hz~)-lhYn4
(x', zJ) for (x, z) with (x~, z~) for X n , Zn such that the matrices A, B, C for eqns.(7)-(9) are diffeomorphically transformed into the form A' , B',Cl respectively, along the equilibrium locus at the neighbourhood of15
Therefore, when a: increases through 6:, one eigenvalue of the system (i.e~, an eigenvalue of J evaluated along the equilibrium locus) moves from less
{
gv~
-
* is an appropriate matrix with each element having orem 5. 4, then there exists a local coordinates the form of constant + O(Q - a)~
A'
(~+o(a:-a)
-
c\a-~J
-
B'
c(~~~a) + CJ(o: - a) C + 0(0 - a)
=:: (
0
CZn,
Zn
C1
0
_(
-
+ O(n -
C Y1 ,n
than modulus 1 if bzlc < 0) by diverging through ±oo. The other (n x + n y - 1) eigenvalues remain bOWlded, change in the order of O(a - a) and stay close to those of Cl due to bz i= 0, c ~ 0 and
0 ) ex +O(a-a) ,
0
C1J~
a)
)
+ 0(0 -
a)
IlbzU > Hbznll.
,
0a))
+ 0(0: -
where x' == x'(x,xn,Yn,z,Zn,Q) E Rnz,z' z'(x,xn,Yn,z,Zn,ll') E Rnz are Coo mappings
in the open neighbourhoods of jJ and C x is a nonsingular matrix. y~ = {Y2,n, ... , Yn~,n}. ex, CZn.' CXn ' CY1,n. and Cy~ are (n x - 1) x (n x 1), (n;z: -1) x 1, (n x -1) x (n x -1), (n x -1) x 1 and (n x - 1) x (n y - 1) matrices respectively, which are all independent of (a - a). Proof of Theorem 5.4. ''''le first show that the equilibrium locus is transversal to the singular surface. Since the matrix of eqn.(18) is nODsingular, there exists an unique equilibrium locus (x(a), y(a), z(a:), et) E R.n~+nll+nz+l locally near Q = Q. Then along the equilibrium locus at a, c~
Jz
+ f~n +
9$n. h~n
i'lln
9Yn h Yn
f%
E
h z
+ fZ n
+
e (
c(:~Q) + 0(0 o
go. ha
hzt'),
0
Ci) e TCz
)
+ CJ(a: -
-
cQc.;lc.t::n
c O C; l
(
1)
+
C '1tl •
+ 0(01
0(0 -
~+
-
a)
0
coC.;l CZTJ
_
O(CIl _ a)
COC;lC 1
~ lJv n
Example 5.1
(nr
+ 0(0
-
a}
where
)
tin
0(0 -
&)
..
COC~lCx.,...
Wlstable manifold and one dimension invariant sin-
0
b~t:%D" c(:~"') _- ~ +
_ ( Cl -
It can be shown from Theorem 5~ 4 that the dynalnics can be locally decomposed into nB-dimension invariant stable manifold, a nU-dimension invariant
0(0 ~ 0)
respectively, due to Lemma 5. 1, where Co = E. Hence at the transformed coordinates (x',x~,Yn,z',z~), the Jacobian matrix J for (x~, Yn) has the form J',
where
while the other (n x + n y · - 1) eigenvalues remain bounded and change in the order ofO(et - et).
)
+
eTC:z; -
= (
• from less than -1 (or -00) to less than modulus 1 i.f bz+b zn < 0 and ~ < O· 'J bz; C 1
Yn+l
a)
a}
I
c'
gular manifold near p where n S and nU be the numbers of stable and unstable bounded eigenvalues of J[5, 6].
according to Lemma 5.1, and furthermore (erA E)A- 1 B and (e'TA - E)A-1C have the forms
~ (e c(::l'a)
• from more than 1 (or 00) to less than modulus 1 zf bz+b zn > 0 and ~ < o·
d erlVe ~ cl
fa)
) -1 (
Slzn.
transformed into
JI
• from less than mod'Ulus 1 to less than -1 by diverging through - 0 0 if b~t~%n < 0 & ~ > 0;
bz
gt!lar surface 1h z I = O. Without loss of generality, next we take the local coordinates (x', z ) with lX~, z~) of Lemma 5~ 1 at p. Then at the new local coordinates, eTA can be
a.nd
• from less than modulus 1 to morn than 1 by diverging through 00 if b:r:t~%n > 0 & ~ > 0;
(19) (20)
0
from eqns.(15)-(17) and eqn4(18) is substituted into eqn.(19). In eqn.(20), Schur's Theorem is also used from the assumptions. Therefore, eqn.. (20) means that the equilibrium locus is transversal to the sin-
(
Remark 5. 1 Assume the same conditions of Theorem 5.4- . When a increases through 0:, one eigenvalue of J evaluated along the equilibrium locus moves
(~,~. ~)T
\vhere h~
~, ~~)T
Zn ) ( ; : ,
#
(
0
Assuming the same conditions as Theorem 5.4 but with IJb z 11 ::; Ilbz 11, it is evident that the singularity-2 induced bir"u.:cations still occur but are degenerate according the proof of Theorem 5. 4 . According to Theorem 5.4 and the Jacobian J' of eqn.(21), we have the following rernark~
£;-6(x(Q), x(o:), yea), zea), z(Q), et)
+ (5x + oZn' 6yn , 5z + C
00.
-
>0 (respectively, from more than modulus 1 to less
than modulus 1 to more than modulus 1 if bz/c
COC;-101l1 .,...
Cl
+
)
o~o: ~ a)
COC;;lC1l £S.B. n 8Yn
) ,
Z~
Zn
~
x
=
= 0'2 - Z + Yn - a O~5Yn + X n - 0:4 + 0.50:
(22) (23)
x - z2 = 0 t < (n + 1)1"; n == 0, 1, 2, .~.)
Lt is a bifurcation parameter, and x, and a E 'R 1 •
(24) Xn ,
Yn,
Obviously x = xn = 0 4 , Yn == a: and Z == zn = 0'2 are a strict equilibrium of eqns.(22)-(24) for any a E 'RI. Let a = 0 and p = (0,0,0,0,0,0), (x(a),y(a),z(a),a) = (a 4 ,a,a 2 ,a) is an wlique (21~quilibliwn locus in n 4 locally near et = O. It is easy to check that eqns~(22)-(24) at p satisfies the conditions of Theorem 5. 4 which means that the singularity-2 induced bifurcation occurs at p when et changes.
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NONLINEAR ANALYSIS OF SAMPLED-DATA CONTROL SYSTEMS...
5.2
Intersampling instants
In this section, we will show that all of the bifurcations except a "degenerate" singularity induced bifurcation for a strict equilibrium at the intersampling instants are related to those at the sampling .instants. In other words, there is no independently local bifurcations at the intersampling instants, except the singularity induced bifurcations which include both independent and dependent bifurcations on those of the sampling instants. The stability of the DDA is determined by J which is the Jacobian of the equivalent discrete system for eqns.(1)-(3) according to Theorem 3. 2 ~ Therefore, there is no biflll'cation at the intersampling instants as far as 1h z I =f. 0 at p. Although the stability of eqns.(1)-(3) is reined by J, the trajectory of (xCt), z(t)) at the intersam-
pling instants follows eqns.(l) and (3) which is a differential-algebraic model.
Theorem 5.5 Suppose that 15 = (x,x,y,z,z,a) is a strict equilibrium of eqns,,(1)-(3) with onedimensional paramete.r 0: E R~ Assume that lhzl p has an algebmically simple zero eigenvalue and bz = -trace { fzadj(hz)h x } is nonzero, there is no purely imaginary eigenvalues for Jacobian matrix A and furthermore the following conditions are satisfied, Ix fz fa and h x h z ha: #- o.
{: I~o,
6x
8z
Ca
Then there exists a smooth curve of equilibria (of eqns.(l) and (3)) in R n :r+n z+ 1 which passes through 15 and is tmnsversal to the singular surface at jJ. When Q increases through a, one eigenvalue of the system for (x,z) ofeqns~(l)-(3) (i.e., an eigenvalue of A evaluated along the equilibrium locus) moves from c- to C+ if bz/c > 0 (respectively, from C+ to if bz/c < 0) along the real axis by diverging through oo~ The other (n x - 1) eigenvalues remain bounded and change in the order of 0(0'-0), where
c-
C
= Oa - (oz, 6z )
(~:
{:) -1
(
{: )
evaluatOO
at p. We call the bifurcations depicted in Theorem 5. 5 as singularity-l induced bifurcations which occur at intersampling instants. Remark 5. 2 All of the local bifurcations except singularity-induced bifurcations at the intersampling instants are related with those of the sampling in.sta·nts.
5.3
Genericity of local bifurcations
This section will show that the DDA, eqns. (1)-(3) has five generic bifurcations described in Theorems 5. 1 -5. 5 under certain generic assumptions. We first give several definitions for structurally stability and bifurcation set [1]. Definition 5. 1 A diffeomorphism map or vector field F(x) : X ---+ X E n m is called structurally stable in X if there is an t: > 0 such that whenever sUPzEx{IIF(x) - G(x)11 + IIFz(x) - Gx(x)11 < E} for G : X ---+ X) F is topologically equivalent (conjugate) to G] in other words, there exists homeomorphism H = X --+ X such that Ho F == Go H. De:finition 5. 2 The bifurcation set is the set of po~nts for which the dynamics is structumlly unstable under variation of pammeters~
14th World Congress of IFAC
Let EP = {x n , Yn, Zn, et} and BP = {x n , yn.~ i~, a} denote the set of strict equilibria and the bifur-
cation set of the strict equilibria for eqns~(1)-(3),
respectively. Therefore EP = {(x n , Yn, Zn, a) : f(p) == 0, Yn == g(p), h(P) = 0 where p == (x n , X n , Un, Zn, Zn, o:)} which is clearly an open subset of nn:c+ny+nz:+l. Define the Hurwitz matrix H rn - 1 as al
a3
a2
a5 a4
a2m.-3
ao 0 0
al
aa
a2m-5
a2
a4
a2m.-6
o
0
0
am-l
a21n-4
(25)
and H rn - 1 (AE-A) means that the elements
ai
cor-
respond to the efficients of polynomial lAE - AI == aoA7'n+ a1 .A. Tn - 1 + ~.~ + am. for m x m matrix A, while Hm-l(~E-A) means that the elements ai correspond to the efficients of I ~=~ E - AI == (ao..\Tn + atAtn-1 + ... + am.)!(l - A)1n-l for m x m matrix A. IH~-l(AE - A)I ~ 0 implies that A has purely imaginaryeigenvalueswhile IHm-l(~!~E-A)i= 0 means that A has modulus 1 eigenvalues. Then we directly have the following lemma. Lemma 5. 2 The bifurcation points for strict equilibria of eqns~(1)-(3) are composed by four sets BP = BP n (SSI U 8PD U SFP U SNS) (26) where SSI = 8FD
= O}
{(:r:n.. Yn~.a:n, Q) E EP : Ihzl {(:Xn. 1.'n., Zn. 0) E EP : IJ - EJ {(~nl Un. %nt 0:) E BP: IJ El
=
8FP
=
SNS
=
+
{(:Z:ndln; zn.O:) E EP:
=
D}
= O}
IHna:+nJil-l(~E- J)I =
(27) (28)
(29) O}.(30)
Notice that the system does not generally undergo bifurcations at the set {(x n , Yn, Zn, a) E EP : 1hz + h Zn I = O}, because one of the bifurcations in Theorems 5. 1- 5.4 will occur before 1h z + h zn I
approaches zero~ It is evident that each set in Lemma 5. 2 is restricted by only one equality on EP~ Let Ssil, Std, Sip, Sns and Ssi2 be the sets of points satisfying the conditions of 5. 5 ,5. 1 ,5. 2 ,5. 3 and 5. 4, respectively. Then, we suppose the following generic assumptions. Assumption 5. 2 Ssil, S Id, Sun Sns and Ssi2 are open and dense in SSI, SFD, ~FP, SNS and SSI, respectively~
The generic bifurcations of eqns.(l)-(9) are constituted by the bifurcations of Theorems 5. 5 ,5~ 1 ,5. 2 ,5~ 9 and 5~ 4 under Assumption 5.2. Proof of TheorelIl 5. 6 . Since only additional independent non-equalities are restricted on 8 s i1, Bid, SIp, SnB frOIIl 8S1, SFD, SFP, SNS, due to Assumption 5. 2 we get Theorem 5. 6
Bid = SPD,S/p:=::: SFp,Sns ~ SNS,S8il = SSI (31) where S means the closure of a set S. Moreover compared with Ssil, Ssi2 is also constrained by an inequality Jlbzfl > IJbzn. II in addition to several nonequalities from SSI. Hence 8 si2 C Ssil. Considering that Ssi2 is also a dense and open subset of SSI according to Assumption 5.2, we express BSI by B sil U Ssi2 = BSI. Therefore according to Lemma 5. 2 , we have BP == BP n (Bsil U Ssi2 U Bid U SJp U Sns) (32)
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NONLINEAR ANALYSIS OF SAMPLED-DATA CONTROL SYSTEMS...
theorern~ 0 Singularity-l induced bifurcation can generally be
which proves this
expected to occur when the system undergoes the singuiarity-2 induced bifurcation due to Ssi.2 C Bsil , although they are restricted by different inequalities and non-equality, Le~ Ssi2 is not a subset of Sail. However, most of the independently locally bifurcations at the intersampling instants are composed of the degenerate singularity-l induced bifurcations as shown in the proof of Theorem 5. 5 , according to the genericity from Theorem 5. 6 . Remark 5. 3 Most of locally nondegenerate bifurcations for the DDA at strict equilibria are related with the local bifurcations of the sampling instants
under Assumption 5.2.
6
Application to Power Systems
We show how a power system with digital PSS can be treated as a DDA in this section. For a power system, the dynamics of the generators and analog controllers(e~g" AVR) is defined as eqn.(l) of the DDA, while eqn.(3) is composed of the power bal-
ance equations. Then eqn.(2) corresponds to the discrete-time controllers (or digital PSS). In this paper, the parameters of the digital PSS are first designed by imitating the conventional analog PSS, and then are adjusted and evaluated by using Theorems 3.. 2 -5. 4 . Next in order to obtain the digital PSS by imitating the designing of analog PSS, we derive eqn. (2) from an analog PSS with a hold ciruit and a sampler. Generally, this process is not necessary if the digital PSS or eqn.(2) is given. Generally, the digital PSS consists of a sampler, a discrete-time controller and a hold circuit. In this paper, we adopt three-order digital PSS with a zero-. order hold circuit which is obtained by discretizing the analog PSS~ Le. "Control device". The state y(t) E 1(,3 and input Un E 1l of the PSS follow D1y(t) + D2Un U.(Xn , zn) == u(x(nT), z(nr)) Un == (nT ~ t < (n + l)T; n == 0, 1, ...) (33) and the output V n E 'R of the digital PSS can be expressed as
yet) =
v(t) =
-
{
Vma.:t
if v(t) ~
Vma:r
Vm.in
If v(t) $
Vnt.in
~:~: VI (t)
(nT :S t
+ Ry~(t) + 1/3(t) + ~~~ Un
< (n + 1)7; n
Vn
14th World Congress of IFAC
Yn+l == eD~TYn + D 1 1 [e Dl 7" - E]D2u(xn , zn) (35) which has the same form with eqn,,(2). That is, eqn~(35) is the discrete-time controller whose design imitates that of the continuous-time controller, i.e~ a three-order analog PSS. Note that the real power output u(x n , zn) of the generator is a given function of (x n , zn). The PSS output V n is obviously a f\ll1ction of '})n and u(xn , zn) according to eqn.(34), Le., Vn
7
[
_*';2~~a -
T4 -zT5
2
T4
-
_
4
D2
= [
conditions~
Finally, the theoretical results were ap-
plied to the evaluations of digital control for power systems.
References
}
4t ¥t[a ]. KT;
Conclusion
the system unstable, and derived their bifurcation
are upper and lower
>
(36)
then, this paper showed that there are five types of
== VenT)
* :]'*
T4'1-' T p 4
=== 71(Yn, u(x n , zn))
generic bifurcations at strict equilibria which cause
bOlillds for the internal output of the PSS. Un and V n are the values of u(t) and v(t) at instant t = nT respectively, which are constantly hold during nr ~ t < (n + 1)r. Define D 1 and D 2 as follows
D1=
Zn)
We first described locally sampling, asymptotical stability anq. singular perturbation for DDA. And
= o~ 1, ...),
V'Jna~,.VJl1,in
7Jn (x n , Yn,
where V n is not a linear function because of the saturation Vmin,Vrnax of vet) shown in eqn.(34). On the other hand, the dynamics of generators and analog controllers corresponding to eqn.(l), the power balance equations corresponding to eqn.(3) can be written as follows x(t) = f(x(t), 17n(Xn , Yn, Zn), z(t)) (37) h(x(t), l1n(Xn )Yn) Zn), z(t)) == O~ (38) Notice that X n , Yn and Zn can be viewed as constants during nT :5 t < (n + 1)7 for eqns~(37)-(38) because Un is constantly hold during that period. Eqns.(37)-(38) are generally nonlinear. Therefore, combining eqns~(37)(35)(38),we can use the Theorems 3.2 -5.4 to analyse the power systems with the digital PSS nwnerically in a nonlinear way. Several numerical simulations have verified that our results are useful[S, 6].
(34) where, yet) = (Yl(t)'Y2(t)'Y3(t))T . u E n is the real power output of the generator which is also a given function of (x, z)~ v E 1?, (or vn E 'R) is the PSS output, and
=
T 2_ Ti
~~ 4
Since eqn~(33) is a linear differential equation, it can be analytically solved. Therefore, we have following difference equation by integrating eqn~(34) from fiT to (n + 1)T,
[1] J. G uckenheimer , P"Holmes.(1983). Nonlinear Oscillations, Dynamical Systems, and Bifurca-
tions of Vector
Fields~
Springer- Verlag, New
York.
[2) V.Venkatasubramanian, H.Schattler, J.Zaborszky~ (1992)~ A stability theory for large differential algebraic systems. Report SSM 9201-Part 17 Department of Systems Science and Mathematics, Washington University-, St.Louis,MO. [3] V.Venkatasubramanian, H.Schattler" J.Zaborszky~ (1995). Local Bifurcations and Feasibility Regions in DifferentialAlgebraic Systems. IEEE 'Prans. on AG, 40, No.12, pp~1992-2013. [4] L.Chen, K.Aihara. (1997). Chaos and asymptotical stability in discrete-time neural networks. Physica D, 104, pp.286-325. [5] L.Chen, K.Aihara. (1997). Local Stability in Hybrid Dynamical Systems. 1997 International Symposium on Nonlinear Theory and its Applications (NOLTA '97), pp.73-76. [6] L.Chen, K.Aihara. (1998). Stability and Bifurcation Analysis of Hybrid Dynamical Systems. Technical Reports, METR 98-02, The University of Tokyo. [7] R~L.Grossman, A"Nerode, A.P.Ravn, H.. Rishel (edited). (1993). Hybrid Systems. SpringerVerlag, New York..
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